A Version of the Paley-Wiener-Schwartz Theorem in Infinite Dimensions

Abstract

Abstract The Paley-Wiener-Schwartz theorem characterizes the Fourier transforms of distributions with bounded (compact) support as being exactly the entire functions of exponential type which are slowly increasing (cf.[4], [18], [20], [2l]). Nachbin and Dineen [9] defined the Frechet space ɛNbc (E;F) of infinitely nuclearly differentiable mappings of boundedcompact type from E valued in F, when E is a real Banach space and F is any Banach space (§1). When E is finite dimensional and F = C, the space ɛNbc (E;C) = = ɛNbc (E) is the space ɛ(E) endowed with the Schwartz topology [20]. For this reason and on account of theorem 3, ɛ'Nbc (E), the dual space to ɛNbc (E), is called the space of distribution with bounded support in infinite dimensions. In contrast with the finite dimensional case, if E is infinite dimensional, then there exist complex valued holomorphic functions of exponential type on (E')c, bounded on E' (and hence slowly increasing) which are not the Fourier transform of any distributions with bounded support (cf. [9]). Here I establish, as a main result of this work, a necessary and sufficient condition for a complex valued holomorphic function of exponential type on (E') c and slowly increasing on E' (when E belongs to a wide class of separable Banach spaces) to be the Fourier transform of a distribution with bounded support: the Paley-Wiener-Schwartz theorem in infinite dimensions.